Polkinghorne’s Meaning in Maths

John Polkinghorne (Ed.) Meaning in Mathematics OUP 2011

In what sense do numbers ‘exist’? What do we mean when we assert a mathematical theorem? Timothy Gowers and Marcus du Sautoy give their views as mathematicians, Polkinghorne and Penrose give their views as mathematical physicists, while some leading British philosophers summarise and extend some metaphysical views.

I am on the look-out for material that could help explain what mathematics is and is not to non-mathematical policy-makers, with a view to avoiding the misunderstandings that characterised the financial crises of 2007/8, for example. I do not think this book at all accessible and nor do I think that what its contributors are trying to say is particularly helpful.

The philosophers note the view by some that all ‘knowledge’ is a product of the brain, and hence questionable. Some give a special place to empirical ‘knowledge’ of the ‘physical world’, and regard this as more ‘objective’ than mathematics. It seems to me that our ‘knowledge’ of the ‘physical world’ really is of a different kind from our knowledge of mathematics, but that both rest on logic. For example, the number ‘3’ has a different kind of existence to the sun. Much of the debate, it seems to me, hinges around the idea that the sun is a ‘first class’ physical object whereas ‘3’ is only ‘second class’. I agree that they are different, but not with the implication that the sun is a superior kind of thing to ‘3’. (If I am in a coma then the sun is a figment of my imagination, whereas ‘3’ is no less real.)

Further, it seems to me that mathematical knowledge relies on less logic and less sensation (if any) than physical ‘knowledge’, and so it is at least as well ‘grounded’, and so it is hard to see how it could be of an inferior status, apart from the difference of subject.

It seems to me that basic arithmetic (such as ‘2+3=5’) is ‘true’ in the strongest possible sense: that if I found some apparent counter-example I would have to question everything that I thought I knew, mathematical and physical. I think that any sufficiently advanced civilization would have to have something corresponding to basic arithmetic. I may be wrong in this, but if I am then everything that I think I know will be in doubt – at least for a while. (I may recover.)

A millennium ago, all mathematicians thought that Euclid’s Geometry was ‘good mathematics’. After quite a few centuries a flaw was found. Some hold that ‘mathematics’ is whatever mathematicians think it is, so here is a case where ‘mathematical knowledge’ was false. But it seems to me that there is some idealised sense in which Euclid was always wrong and his Geometry was only ‘thought to be’ mathematics, it was not really. Thus – I think – we can say that mathematics is always correct, but that mathematicians can be in error about what is really mathematics. Alternatively, we might say that mathematics is relative to some standard of logic, so that if our standard changes (as it did about 100 years ago) then what we mean by mathematics changes. We can admit the possibility that we might need to refine our logic and hence that our mathematics is ‘only’ as good as we know how to make it while recognizing that it is based on the absolute minimum of perceptions and assumptions and hence ‘as good as it gets’ – as long as we really do expose our logic to criticism.

A further feature of Geometry is that most (perhaps all) people, including mathematicians, thought that its axioms were literally true of the physical world. But this never was a mathematical belief: it was only ever a physical belief. Thus while mathematicians were mistaken, it was not a mathematical mistake. It was something that could be, and was, corrected by science. (Mathematics can never be contradicted by any science, unless it be very bad science.)

Is this just metaphysical? The mathematical theory of probability is widely applied to finance and economics. What is the justification? In so far as finance and economics are about the real world, mathematics can say nothing about the appropriateness of probability theory. Some sort of scientific argument would be required. Here one needs to understand the ‘meaning’ of science. But it does seem to be the case that prior to 2006 there was no evidence that there would be a crash, and that Occam’s razor (a principle often applied within sciences) would have it that no ‘scientific’ theory would have predicted a crash in 2008. But this does not preclude the possibility that if economists had looked harder for evidence and not been so confident in their models, they might have seen some warning signs – as some have since claimed to do. But my key point is that it ought to have been obvious that statements such as ‘a crash is logically, mathematically or statistically impossible’ were bogus. We need some exposition that helps policy makers to understand this.